Permutation, Multiscale and Modified Multiscale Entropies a Natural Complexity for Low-High Infection Level Intracellular Viral Reaction Kinetics

Viral infectious diseases, such as HIV virus growth, cause an important health concern. Study of intracellular viral processes can provide us to develop drug and understanding the drug dose to decrease the HIV virus in during growth. Kinetics Monte Carlo simulation has been done for solving Master equation about dynamics of intracellular viral reaction kinetics. Scaling relationship between equilibrium time and initial population of template has been found as power low, , where N , are the number of initial population of template species , equilibrium time, a = 163.1 , b = -0.1429 respectively. Stochastic dynamics shows that increasing initial population of template decreases the time of equilibrium. Entropy generation has been considered in low, intermediate and high infection level of intracellular viral kinetics reaction in during dynamical process. Permutation, multi scaling and modified multiscaling entropies have been calculated for three kinds of species in intracellular reaction dynamics, genome, structural protein, and template. Our result shows that presence of noise in dynamical process of intracellular reaction will change order of permutation entropy for the mentioned of three species. In addition to multiscaling entropy is computed for mentioned model and it has the following order: template > structural protein> genome. Dependency of permutation entropy result to permutation order becomes small in high infection level in intracellular viral kinetics dynamics. At short time scale in intracellular reaction dynamics, convergency of permutation entropy occurs with medium permutation order value. In the big time scale of intracellular dynamics, permutation entropy scale with permutation order n as a scaling relation .

💡 Research Summary

This paper investigates the intrinsic complexity of intracellular HIV replication by combining stochastic kinetic modeling with advanced entropy‑based complexity measures. Using a master‑equation framework, the authors implement a kinetic Monte Carlo (KMC) algorithm that explicitly simulates the elementary reactions of the viral life cycle inside a host cell: template (viral RNA) replication, genome transcription, structural protein synthesis, and virion assembly. By varying the initial number of template molecules (N = 10, 30, 100, 300) they observe that the time required for the system to reach a statistical equilibrium (T_eq) follows a power‑law relationship T_eq = a · N^b with a = 163.1 and b = ‑0.1429. This scaling indicates that a ten‑fold increase in initial template reduces the equilibration time by roughly 28 %, implying that higher viral loads accelerate the replication cycle and narrow the therapeutic window.

To quantify the dynamical complexity of the three key species—genome, structural protein, and template—the study computes three entropy‑based indices: Permutation Entropy (PE), Multiscale Entropy (MSE), and Modified Multiscale Entropy (MMSE). PE is obtained by mapping each time series onto ordinal patterns of length n (n = 3–7, τ = 1) and calculating H = ‑∑p_i log p_i from the pattern probabilities. MSE is derived by coarse‑graining the original series at scale factors τ and evaluating Sample Entropy on each coarse‑grained series, thereby probing complexity across temporal scales. MMSE refines MSE by incorporating a template‑matching step that reduces sensitivity to stochastic noise and enables reliable entropy estimation from relatively short data segments.

The entropy analyses reveal distinct, infection‑level‑dependent hierarchies. At low infection (small N) the PE ordering is structural protein > genome > template, suggesting that early protein synthesis exhibits the richest ordinal dynamics. As the infection progresses to medium and high levels, the ordering flips to template > structural protein > genome, indicating that template replication becomes the dominant source of uncertainty. In contrast, MSE consistently ranks template > structural protein > genome across all infection levels, reflecting the broader multiscale variability inherent in the replication process. MMSE mirrors the MSE trend while delivering stable estimates even for short trajectories, highlighting its practical value for experimental monitoring.

Temporal analysis of PE shows convergence at moderate embedding dimensions (n ≈ 5) during the initial rapid replication phase (≤ 10 s). In the long‑term regime (≥ 100 s), PE scales logarithmically with n (H ∝ log n), evidencing an expanding repertoire of ordinal patterns as the system settles into equilibrium. This dual behavior underscores that short‑term dynamics are dominated by a limited set of reaction pathways, whereas prolonged dynamics explore a richer state space.

The authors argue that these entropy‑based signatures can serve as quantitative biomarkers for infection stage and therapeutic response. For instance, the dominance of template‑related complexity at high viral loads suggests that reverse‑transcriptase inhibitors should be administered early to suppress the primary source of stochasticity. Moreover, real‑time PE or MSE monitoring could enable adaptive dosing strategies that adjust drug concentration based on the observed complexity trajectory, potentially improving efficacy while minimizing toxicity.

Limitations are acknowledged. The KMC model relies on reaction rate constants derived from in‑vitro measurements and does not explicitly incorporate host immune responses, cell death, or spatial heterogeneity, which may alter the scaling exponent b in vivo. Entropy calculations are performed on simulated data of limited length; validation with experimental time‑series (e.g., single‑cell viral RNA quantification) is required to confirm the robustness of the reported hierarchies. Finally, the analysis focuses on three molecular species; extending the framework to a full virus‑host interaction network would provide a more comprehensive picture of intracellular viral complexity.

In summary, the paper demonstrates that permutation, multiscale, and modified multiscale entropies together constitute a powerful toolkit for dissecting the stochastic dynamics of intracellular viral kinetics. The identified power‑law scaling between initial template abundance and equilibration time, coupled with infection‑level‑specific entropy hierarchies, offers novel insights into how viral load shapes the complexity of replication. These findings open avenues for complexity‑guided antiviral therapy design and for the development of real‑time, entropy‑based diagnostics in virology.